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We give a gauge-invariant definition of the vortex surface in SU(N) Yang–Mills theory without using the gauge fixing procedure. In this construction, gauge-invariant magnetic monopoles with fractional magnetic charges emerge in the boundary of the non-oriented vortex surface such that the asymptotic string tension reproduces the correct N-ality dependence. We show that gauge-invariant magnetic monopoles and vortices are simultaneously responsible for quark confinement in four-dimensional spacetime based on the Wilson criterion. These results are extracted from a non-Abelian Stokes theorem derived in the previous paper.